Abstract

A theoretical framework is proposed for describing the laser Doppler photodetector signal. The theory allows for predicting the power of the photocurrent fluctuations. It is valid for a detector of arbitrary size. The input data required for application of the theory are the angular distribution of the detected light, the fraction of Doppler-shifted photons, and the active detector size. The theory is based on the time-domain approach to the statistics of dynamic speckle patterns on the photodetector. An experiment was carried out to validate some aspects of our theory. The consequences of the speckle dynamics for the various modes of laser Doppler flowmetry are discussed.

© 2001 Optical Society of America

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References

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  1. J. D. Briers, “Laser Doppler and time-varying speckle: a reconciliation,” J. Opt. Soc. Am. A 13, 345–350 (1996).
    [CrossRef]
  2. R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
    [CrossRef] [PubMed]
  3. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  4. J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
  5. R. Dändliker, “Heterodyne holographic interferometry,” in Progress in Optics XVII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 1–84.
  6. R. Thalmann, R. Dändliker, “Statistical properties of interference phase detection in speckle fields applied to holographic interferometry,” J. Opt. Soc. Am. A 3, 972–981 (1986).
    [CrossRef]
  7. H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1970), Vol. VIII, pp. 133–200.
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).
  9. J. D. Briers, “The statistics of fluctuating speckle patterns produced by a mixture of moving and stationary scatterers,” Opt. Quantum Electron. 10, 364–366 (1978).
    [CrossRef]
  10. A. T. Forrester, R. A. Gudmudsen, P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
    [CrossRef]
  11. T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1995), Vol. XXXIV, pp. 183–248.
  12. F. F. M. de Mul, J. van Spijker, D. van der Plas, J. Greve, J. G. Aarnoudse, T. M. Smits, “Mini laser-Doppler (blood) flow monitor with diode laser source and detection integrated in the probe,” Appl. Opt. 23, 2970–2973 (1984).
    [CrossRef] [PubMed]
  13. W. Steenbergen, M. van Stratum, F. F. M. de Mul, J. Greve, “Coherence effects in laser Doppler blood flow metry,” in Optical Diagnostics of Biological Fluids and Advanced Techniques in Analytical Cytology, A. V. Priezzhev, T. Asakura, R. C. Leif, A. Katzir, eds., Proc. SPIE2982, 6–17 (1997).
    [CrossRef]
  14. K. Wårdell, A. Jakobsson, G. E. Nilsson, “Laser Doppler perfusion imaging by dynamic light scattering,” IEEE Trans. Biomed. Eng. 40, 309–316 (1993).
    [CrossRef] [PubMed]
  15. T. J. H. Essex, P. O. Byrne, “A laser Doppler scanner for imaging blood flow in skin,” J. Biomed. Eng. 13, 189–193 (1991).
    [CrossRef] [PubMed]
  16. S. J. Matcher, M. Cope, D. T. Delpy, “In vivo measurements of the wavelength dependence of tissue-scattering coefficients between 760 and 900 nm measured with time-resolved spectroscopy,” Appl. Opt. 36, 386–396 (1997).
    [CrossRef] [PubMed]
  17. F. F. M. de Mul, M. H. Koelink, M. L. Kok, P. J. Harmsma, J. Greve, R. Graaff, J. G. Aarnoudse, “Laser Doppler velocimetry and Monte Carlo simulations on models for blood perfusion in tissue,” Appl. Opt. 34, 6595–6611 (1995).
    [CrossRef] [PubMed]

1997

1996

1995

1993

K. Wårdell, A. Jakobsson, G. E. Nilsson, “Laser Doppler perfusion imaging by dynamic light scattering,” IEEE Trans. Biomed. Eng. 40, 309–316 (1993).
[CrossRef] [PubMed]

1991

T. J. H. Essex, P. O. Byrne, “A laser Doppler scanner for imaging blood flow in skin,” J. Biomed. Eng. 13, 189–193 (1991).
[CrossRef] [PubMed]

1986

1984

1981

1978

J. D. Briers, “The statistics of fluctuating speckle patterns produced by a mixture of moving and stationary scatterers,” Opt. Quantum Electron. 10, 364–366 (1978).
[CrossRef]

1955

A. T. Forrester, R. A. Gudmudsen, P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Aarnoudse, J. G.

Asakura, T.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1995), Vol. XXXIV, pp. 183–248.

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Bonner, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

Briers, J. D.

J. D. Briers, “Laser Doppler and time-varying speckle: a reconciliation,” J. Opt. Soc. Am. A 13, 345–350 (1996).
[CrossRef]

J. D. Briers, “The statistics of fluctuating speckle patterns produced by a mixture of moving and stationary scatterers,” Opt. Quantum Electron. 10, 364–366 (1978).
[CrossRef]

Byrne, P. O.

T. J. H. Essex, P. O. Byrne, “A laser Doppler scanner for imaging blood flow in skin,” J. Biomed. Eng. 13, 189–193 (1991).
[CrossRef] [PubMed]

Cope, M.

Cummins, H. Z.

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1970), Vol. VIII, pp. 133–200.

Dainty, J. C.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

Dändliker, R.

R. Thalmann, R. Dändliker, “Statistical properties of interference phase detection in speckle fields applied to holographic interferometry,” J. Opt. Soc. Am. A 3, 972–981 (1986).
[CrossRef]

R. Dändliker, “Heterodyne holographic interferometry,” in Progress in Optics XVII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 1–84.

de Mul, F. F. M.

Delpy, D. T.

Ennos, A. E.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

Essex, T. J. H.

T. J. H. Essex, P. O. Byrne, “A laser Doppler scanner for imaging blood flow in skin,” J. Biomed. Eng. 13, 189–193 (1991).
[CrossRef] [PubMed]

Forrester, A. T.

A. T. Forrester, R. A. Gudmudsen, P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Francon, M.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

Goodman, J. W.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

Graaff, R.

Greve, J.

Gudmudsen, R. A.

A. T. Forrester, R. A. Gudmudsen, P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Harmsma, P. J.

Jakobsson, A.

K. Wårdell, A. Jakobsson, G. E. Nilsson, “Laser Doppler perfusion imaging by dynamic light scattering,” IEEE Trans. Biomed. Eng. 40, 309–316 (1993).
[CrossRef] [PubMed]

Johnson, P. O.

A. T. Forrester, R. A. Gudmudsen, P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Koelink, M. H.

Kok, M. L.

Matcher, S. J.

McKechnie, T. S.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

Nilsson, G. E.

K. Wårdell, A. Jakobsson, G. E. Nilsson, “Laser Doppler perfusion imaging by dynamic light scattering,” IEEE Trans. Biomed. Eng. 40, 309–316 (1993).
[CrossRef] [PubMed]

Nossal, R.

Okamoto, T.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1995), Vol. XXXIV, pp. 183–248.

Parry, G.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Smits, T. M.

Steenbergen, W.

W. Steenbergen, M. van Stratum, F. F. M. de Mul, J. Greve, “Coherence effects in laser Doppler blood flow metry,” in Optical Diagnostics of Biological Fluids and Advanced Techniques in Analytical Cytology, A. V. Priezzhev, T. Asakura, R. C. Leif, A. Katzir, eds., Proc. SPIE2982, 6–17 (1997).
[CrossRef]

Swinney, H. L.

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1970), Vol. VIII, pp. 133–200.

Thalmann, R.

van der Plas, D.

van Spijker, J.

van Stratum, M.

W. Steenbergen, M. van Stratum, F. F. M. de Mul, J. Greve, “Coherence effects in laser Doppler blood flow metry,” in Optical Diagnostics of Biological Fluids and Advanced Techniques in Analytical Cytology, A. V. Priezzhev, T. Asakura, R. C. Leif, A. Katzir, eds., Proc. SPIE2982, 6–17 (1997).
[CrossRef]

Wårdell, K.

K. Wårdell, A. Jakobsson, G. E. Nilsson, “Laser Doppler perfusion imaging by dynamic light scattering,” IEEE Trans. Biomed. Eng. 40, 309–316 (1993).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

Appl. Opt.

IEEE Trans. Biomed. Eng.

K. Wårdell, A. Jakobsson, G. E. Nilsson, “Laser Doppler perfusion imaging by dynamic light scattering,” IEEE Trans. Biomed. Eng. 40, 309–316 (1993).
[CrossRef] [PubMed]

J. Biomed. Eng.

T. J. H. Essex, P. O. Byrne, “A laser Doppler scanner for imaging blood flow in skin,” J. Biomed. Eng. 13, 189–193 (1991).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Quantum Electron.

J. D. Briers, “The statistics of fluctuating speckle patterns produced by a mixture of moving and stationary scatterers,” Opt. Quantum Electron. 10, 364–366 (1978).
[CrossRef]

Phys. Rev.

A. T. Forrester, R. A. Gudmudsen, P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Other

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1995), Vol. XXXIV, pp. 183–248.

W. Steenbergen, M. van Stratum, F. F. M. de Mul, J. Greve, “Coherence effects in laser Doppler blood flow metry,” in Optical Diagnostics of Biological Fluids and Advanced Techniques in Analytical Cytology, A. V. Priezzhev, T. Asakura, R. C. Leif, A. Katzir, eds., Proc. SPIE2982, 6–17 (1997).
[CrossRef]

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

R. Dändliker, “Heterodyne holographic interferometry,” in Progress in Optics XVII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 1–84.

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1970), Vol. VIII, pp. 133–200.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

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Figures (7)

Fig. 1
Fig. 1

Decomposition of the local field complex amplitude into its components, either Doppler shifted (dashed arrows) or unshifted (solid arrows).

Fig. 2
Fig. 2

Illustration of the Zernike–Van Cittert theorem: spatial coherence function generated by an incoherent extended source at a distance Z from a screen.

Fig. 3
Fig. 3

The three most important modes of Laser Doppler blood flowmetry in tissue. LD, laser diode; PD’s photodetectors.

Fig. 4
Fig. 4

Fiber-optic setup for measurements of the signal modulation on the photodetector as a function of the distance Z between the facet of the receiving fiber and the detector.

Fig. 5
Fig. 5

Experimental results: modulation depth of the photodetector signal versus distance between the fiber facet and the active detector area. The experiment was performed on four Intraplipid™ water suspensions at different concentrations.

Fig. 6
Fig. 6

Method for measuring the Doppler fraction f: (a) calibration, (b) determination of the dc scattered-light component, (c) light on the detector containing both static and dynamic components.

Fig. 7
Fig. 7

Modulation depth normalized with N versus the fraction of Doppler-shifted photons: solid curve, theoretically expected behavior [see Eq. (32)]; symbols, experimentally obtained values.

Equations (41)

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Mi=0P(ω)ωidω,
ic(t)=ic+δic(t),
i2=N2ic2+Nδic2,
iac2AcohAdet.
iac2idc2=AcohAdet=1N,
iac2=AcohAdetR2I2.
i(t)=ρ0LI(x, t)dx.
Γii(τ)i(t)i(t+τ)=ρ20L0LI(x, t)I(x˜, t+τ)dxdx˜,
p=x˜+x-L,
q=x˜-x.
Γii(τ)=ρ2-LL-(L-|q|)(L-|q|)I(x, t)I(x˜(p, q), t+τ)×(x, x˜)(p, q)dpdq,
Γii(τ)=ρ2-LL|q|/2(L-|q|/2)I(p, t)I(p+q, t+τ)dpdq.
Γii(τ)=ρ2-LL(L-|q|)ΓII(q, τ)dq=ρ2L-LL1-|q|LΓII(q, τ)dq,
Γii(τ)=R2-L1L1-L2L2L1L2ΓII(Δx1, Δx2;τ)1-|Δx1|L1×1-|Δx2|L2dΔx1dΔx2,
Γii(τ)=2πAdetR20ΓII(Δr, τ)Δr dΔr.
Γii(0)=idc2+iac2.
ΓII(r, τ)=I2+|ΓEE(r, τ)|2,
Etot(x, t)=E0(x)+ED(x, t).
ΓEE(Δx, τ)Etot(x, t)Etot*(x+Δx, t+τ=E0(x)E0*(x+Δx)+ED(x, t)ED*(x+Δx, t+τ).
Etot(x, t)=i=0MEi(x, t).
ΓEE(x, τ)=i=0MΓEEi(x, τ).
γEE(Δx, τ)ΓEE(Δx, τ)ΓEE(0, 0)=i=0MΓEEi(Δx, τ)I=i=0MfiΓEEi(Δx, τ)fiI=i=0MfiγEEi(Δx, τ).
ΓII(Δr, τ)=I2(1+|γEE(Δr, τ)|2)
=I21+i=0MfiγEEi(Δr, τ)2=ΓIIdc(Δr)+ΓIIac(Δr, τ),
ΓIIdc(Δr)=I2[1+f02|γEE0(Δr)|2],
ΓIIac(Δr, τ)=I22i=1Mf0fiγEE0(Δr)γEE*i(Δr, τ)+i=1Mj=1MfifjγEEi(Δr, τ)γEE*j(Δr, τ).
ΓIIac(τ)=2πAdetR20ΓIIac(Δr, τ)ΔrdΔr=2πAdetR2I2×2i=1Mf0fi0γEE0(Δr)γEE*i(Δr, τ)ΔrdΔr+i=1Mj=1Mfifj0γEEi(Δr, τ)γEE*j(Δr, τ)ΔrdΔr.
Acohij=2π0γEEi(Δx, 0)γEE*j(Δx, 0)ΔxdΔx,i, j=0, 1,, M.
iac2Γiiac(τ=0)=AdetR2I22i=1Mf0fiAcoh0i+i=1Mj=1Mfifj Acohij.
γEE(ΔX, ΔY)
=sourceI(x, y)exp-i 2πλZ (xΔX+yΔY)dxdysourceI(x, y)dxdy,
Acoh=Acoh00==Acohij==AcohMM.
fi=1Mfi,
iac2=R2Adet Acoh f(2-f )I2.
iac2idc2=f(2-f ) 1N.
f=1-1-iac2idc2 N1/2.
fm=[m¯(n)]mexp[-m¯(n)]m!.
iac2idc2=1N [1-exp(-2m¯(n))].
γEE(ΔX)=2J1[(2πΔX)/λZ][(2πΔX)/λZ].
Acoh=Z2λ2πws2=λ2Ω,
Acoh=χ Z2λ2ws2,

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